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Theorem isnumbasgrplem2 43093
Description: If the (to be thought of as disjoint, although the proof does not require this) union of a set and its Hartogs number supports a group structure (more generally, a cancellative magma), then the set must be numerable. (Contributed by Stefan O'Rear, 9-Jul-2015.)
Assertion
Ref Expression
isnumbasgrplem2 ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) → 𝑆 ∈ dom card)

Proof of Theorem isnumbasgrplem2
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 basfn 17249 . . 3 Base Fn V
2 ssv 4020 . . 3 Grp ⊆ V
3 fvelimab 6981 . . 3 ((Base Fn V ∧ Grp ⊆ V) → ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) ↔ ∃𝑥 ∈ Grp (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))))
41, 2, 3mp2an 692 . 2 ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) ↔ ∃𝑥 ∈ Grp (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)))
5 harcl 9597 . . . . . 6 (har‘𝑆) ∈ On
6 onenon 9987 . . . . . 6 ((har‘𝑆) ∈ On → (har‘𝑆) ∈ dom card)
75, 6ax-mp 5 . . . . 5 (har‘𝑆) ∈ dom card
8 xpnum 9989 . . . . 5 (((har‘𝑆) ∈ dom card ∧ (har‘𝑆) ∈ dom card) → ((har‘𝑆) × (har‘𝑆)) ∈ dom card)
97, 7, 8mp2an 692 . . . 4 ((har‘𝑆) × (har‘𝑆)) ∈ dom card
10 ssun1 4188 . . . . . . . 8 𝑆 ⊆ (𝑆 ∪ (har‘𝑆))
11 simpr 484 . . . . . . . 8 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)))
1210, 11sseqtrrid 4049 . . . . . . 7 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ⊆ (Base‘𝑥))
13 fvex 6920 . . . . . . . 8 (Base‘𝑥) ∈ V
1413ssex 5327 . . . . . . 7 (𝑆 ⊆ (Base‘𝑥) → 𝑆 ∈ V)
1512, 14syl 17 . . . . . 6 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ∈ V)
167a1i 11 . . . . . 6 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → (har‘𝑆) ∈ dom card)
17 simp1l 1196 . . . . . . . 8 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → 𝑥 ∈ Grp)
18123ad2ant1 1132 . . . . . . . . 9 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → 𝑆 ⊆ (Base‘𝑥))
19 simp2 1136 . . . . . . . . 9 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → 𝑎𝑆)
2018, 19sseldd 3996 . . . . . . . 8 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → 𝑎 ∈ (Base‘𝑥))
21 ssun2 4189 . . . . . . . . . . 11 (har‘𝑆) ⊆ (𝑆 ∪ (har‘𝑆))
2221, 11sseqtrrid 4049 . . . . . . . . . 10 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → (har‘𝑆) ⊆ (Base‘𝑥))
23223ad2ant1 1132 . . . . . . . . 9 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → (har‘𝑆) ⊆ (Base‘𝑥))
24 simp3 1137 . . . . . . . . 9 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → 𝑐 ∈ (har‘𝑆))
2523, 24sseldd 3996 . . . . . . . 8 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → 𝑐 ∈ (Base‘𝑥))
26 eqid 2735 . . . . . . . . 9 (Base‘𝑥) = (Base‘𝑥)
27 eqid 2735 . . . . . . . . 9 (+g𝑥) = (+g𝑥)
2826, 27grpcl 18972 . . . . . . . 8 ((𝑥 ∈ Grp ∧ 𝑎 ∈ (Base‘𝑥) ∧ 𝑐 ∈ (Base‘𝑥)) → (𝑎(+g𝑥)𝑐) ∈ (Base‘𝑥))
2917, 20, 25, 28syl3anc 1370 . . . . . . 7 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → (𝑎(+g𝑥)𝑐) ∈ (Base‘𝑥))
30 simp1r 1197 . . . . . . 7 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)))
3129, 30eleqtrd 2841 . . . . . 6 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → (𝑎(+g𝑥)𝑐) ∈ (𝑆 ∪ (har‘𝑆)))
32 simplll 775 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑥 ∈ Grp)
3322ad2antrr 726 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → (har‘𝑆) ⊆ (Base‘𝑥))
34 simprl 771 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑐 ∈ (har‘𝑆))
3533, 34sseldd 3996 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑐 ∈ (Base‘𝑥))
36 simprr 773 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑑 ∈ (har‘𝑆))
3733, 36sseldd 3996 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑑 ∈ (Base‘𝑥))
3812ad2antrr 726 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑆 ⊆ (Base‘𝑥))
39 simplr 769 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑎𝑆)
4038, 39sseldd 3996 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑎 ∈ (Base‘𝑥))
4126, 27grplcan 19031 . . . . . . 7 ((𝑥 ∈ Grp ∧ (𝑐 ∈ (Base‘𝑥) ∧ 𝑑 ∈ (Base‘𝑥) ∧ 𝑎 ∈ (Base‘𝑥))) → ((𝑎(+g𝑥)𝑐) = (𝑎(+g𝑥)𝑑) ↔ 𝑐 = 𝑑))
4232, 35, 37, 40, 41syl13anc 1371 . . . . . 6 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → ((𝑎(+g𝑥)𝑐) = (𝑎(+g𝑥)𝑑) ↔ 𝑐 = 𝑑))
43 simplll 775 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑥 ∈ Grp)
4412ad2antrr 726 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑆 ⊆ (Base‘𝑥))
45 simprr 773 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑑𝑆)
4644, 45sseldd 3996 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑑 ∈ (Base‘𝑥))
47 simprl 771 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑎𝑆)
4844, 47sseldd 3996 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑎 ∈ (Base‘𝑥))
4922ad2antrr 726 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → (har‘𝑆) ⊆ (Base‘𝑥))
50 simplr 769 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑏 ∈ (har‘𝑆))
5149, 50sseldd 3996 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑏 ∈ (Base‘𝑥))
5226, 27grprcan 19004 . . . . . . 7 ((𝑥 ∈ Grp ∧ (𝑑 ∈ (Base‘𝑥) ∧ 𝑎 ∈ (Base‘𝑥) ∧ 𝑏 ∈ (Base‘𝑥))) → ((𝑑(+g𝑥)𝑏) = (𝑎(+g𝑥)𝑏) ↔ 𝑑 = 𝑎))
5343, 46, 48, 51, 52syl13anc 1371 . . . . . 6 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → ((𝑑(+g𝑥)𝑏) = (𝑎(+g𝑥)𝑏) ↔ 𝑑 = 𝑎))
54 harndom 9600 . . . . . . 7 ¬ (har‘𝑆) ≼ 𝑆
5554a1i 11 . . . . . 6 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → ¬ (har‘𝑆) ≼ 𝑆)
5615, 16, 16, 31, 42, 53, 55unxpwdom3 43084 . . . . 5 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆* ((har‘𝑆) × (har‘𝑆)))
57 wdomnumr 10102 . . . . . 6 (((har‘𝑆) × (har‘𝑆)) ∈ dom card → (𝑆* ((har‘𝑆) × (har‘𝑆)) ↔ 𝑆 ≼ ((har‘𝑆) × (har‘𝑆))))
589, 57ax-mp 5 . . . . 5 (𝑆* ((har‘𝑆) × (har‘𝑆)) ↔ 𝑆 ≼ ((har‘𝑆) × (har‘𝑆)))
5956, 58sylib 218 . . . 4 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ≼ ((har‘𝑆) × (har‘𝑆)))
60 numdom 10076 . . . 4 ((((har‘𝑆) × (har‘𝑆)) ∈ dom card ∧ 𝑆 ≼ ((har‘𝑆) × (har‘𝑆))) → 𝑆 ∈ dom card)
619, 59, 60sylancr 587 . . 3 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ∈ dom card)
6261rexlimiva 3145 . 2 (∃𝑥 ∈ Grp (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)) → 𝑆 ∈ dom card)
634, 62sylbi 217 1 ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) → 𝑆 ∈ dom card)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wrex 3068  Vcvv 3478  cun 3961  wss 3963   class class class wbr 5148   × cxp 5687  dom cdm 5689  cima 5692  Oncon0 6386   Fn wfn 6558  cfv 6563  (class class class)co 7431  cdom 8982  harchar 9594  * cwdom 9602  cardccrd 9973  Basecbs 17245  +gcplusg 17298  Grpcgrp 18964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-1cn 11211  ax-addcl 11213
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-omul 8510  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-oi 9548  df-har 9595  df-wdom 9603  df-card 9977  df-acn 9980  df-nn 12265  df-slot 17216  df-ndx 17228  df-base 17246  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968
This theorem is referenced by:  isnumbasabl  43095  isnumbasgrp  43096
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